Design and development of non-magnetic hierarchical actuator powered by shape memory alloy based bipennate muscle

Actuators are ubiquitous to generate controlled motion through the application of suitable excitation force or torque to perform various operations in manufacturing and industrial automation. The demands placed on faster, smaller, and efficient actuators drive innovation in actuator development. Shape memory alloy (SMA) based actuators have multiple advantages over conventional actuators, including high power-to-weight ratio. This paper integrates the advantages of pennate muscle of a biological system and the unique properties of SMA to develop SMA-based bipennate actuator. The present study explores and expands on the previous SMA actuators by developing the mathematical model of the new actuator based on the bipennate arrangement of the SMA wires and experimentally validating it. The new actuator is found to deliver at least five times higher actuation forces (up to 150 N) in comparison to the reported SMA-based actuators. The corresponding weight reduction is about 67%. The results from the sensitivity analysis of the mathematical model facilitates customization of the design parameters and understanding critical parameters. This study further introduces an Nth level hierarchical actuator that can be deployed for further amplification of actuation forces. The SMA-based bipennate muscle actuator has broad applications ranging from building automation controls to precise drug delivery systems.

The SMA wire exhibits the shape-memory phenomenon that occurs in the nickel-titanium alloy family. The concentration of the elements is a critical variable in an alloy comprising two or more elements. The presence of nickel and titanium atoms in the alloy is almost in equal ratio 1:1, resulting in the formation of a crystal structure that can transform from one form to another. The temperature at which such phase transformation occurs depends upon the exact composition of the alloy. In the austenite phase, which is present above the transformation temperature, the material shows high strength and cannot be easily deformed under load. The behavior of the alloy is similar to stainless steel; thus, it has the ability to withstand higher stress upon actuation. Generally, SMAs exhibit two temperature-dependent phases, the low and the high-temperature phases. Both the phases have unique properties due to the presence of different crystal structures. The low-temperature phase known as martensite (M) exhibits monoclinic, orthorhombic, or tetragonal crystal arrangements. On the other hand, the high-temperature phase called austenite (A) has a cubic crystal structure. The shear lattice distortion results in the transformation from one phase to another phase. This phenomenon is known as martensitic transformation. Such forward and reverse phase transformations form the base for the unique behavior of SMAs.
The Shape Memory Effect (SME) phenomenon is responsible for the macroscopic shape change of the alloy and its complete recovery during phase transformation. 2 Under zero load condition, when the temperature is at the martensitic start temperature (M s ), austenite begins to transform to twinned martensite and completes the transformation to martensite at the martensitic finish temperature (M f ); this is known as forward transformation. The transformation is complete at this stage, and the material is entirely in the twinned martensite phase. In an equivalent manner, the reverse transformation occurs during the heating process; it starts at the austenitic start temperature (A s ) and gets completed at the austenitic finish temperature (A f ). Under the applied stress condition, the low-temperature twinned martensitic phase is converted to detwinned martensite due to reorientation. The macroscopic shape change is a result of the detwinning process, retaining the deformed configuration when the stress is relieved. Subsequent heating of the SMA to a temperature above A s results in a change of phase from detwinned martensite to austenite (reverse phase transformation), leading to complete shape recovery. When the temperature falls below M s , twinned martensite formation occurs without any shape change observed during forward transformation.

Market access information
According to Fortune Business Insights, the global market size of actuators was $39.08 billion in 2020 and is estimated to reach $91.05 billion by 2028 with a CAGR of 12.06% during 2021-2028 period 3 . Furthermore, as per a report by Markets and Markets, the actuators market is expected to grow from $53.9 billion in 2021 to $86.6 billion by 2027 at a CAGR of 8.2% 4 . The actuator market is segmented on the type of actuator, basis of actuation, vertical, region, and application. The increased investments in oil and gas, power, chemicals, and the rising adoption of international safety standards and practices are expected to influence actuator market growth in the Asia-Pacific region. The actuator market is forecasted to rise at a healthy rate, attributed to an increase in automated systems in industrial applications. With increased focus towards automation of production operations for increased precision, enhanced safety, and higher efficiency is expected to fuel market growth. The rising demand for advanced actuator products is compelling OEMs to invest in R&D activities. Table S1 shows that SMA-based actuators available in the open literature reported actuation force lesser than 30 N with the power consumption at par or more significant than reported in the submitted manuscript [5][6][7] . The parallel arrangement-based linear actuators reported the SMA wire length in the range of 350 mm, and hence its overall envelop packaging exceeds 350 mm 6,7 . The proof-of-concept bipennate-based SMA actuator utilizes 1000 mm of SMA wire for actuation; however, by virtue of the bipennate arrangement, the SMA wires are accommodated within the overall dimensions of 390 mm x 95 mm x 15 mm using an extension type bias spring. In another embodiment with a compression bias spring, a similar bipennate-based linear actuator can be confined within 280 mm length. As evident from Table S1, on average, the net actuation force produced by the bipennate-based SMA actuator is four times more than other SMA-based actuator designs with half the power consumption. The force produced in a single fiber causes macro-level muscle force generation in bipennate musculature. Thus, the current system also gives the functionality of adding even more unipennate SMA branches in the stipulated length. Hence, an even higher force can be generated in the same actuator dimension.

Comparison with existing SMA-based actuators
On the other hand, while the SMA-based actuators available in the literature reported a much higher force, as shown in Table S1, they had significantly higher power consumption than the SMA-based linear actuation system with a bipennate configuration. Upon comparison, Mosley et al. (1999) 8 had a parallel configuration of 48 SMA wire bundles, with approximately 15 times the SMA length and close to 11 times the power supplied, to yield a force approx. 4.5 times the force generated by the current bipennate based SMA actuator. Similarly, Shin et al. (2005) 9 required approximately 150 times the power supplied to yield a force closer to 2 times the force reported in the submitted manuscript, as shown in Table S1. The present bio-inspired shape memory alloy-based hierarchical actuator produces a maximum measured force of 105 N when actuated by a power supply of 7 V pulse voltage with a current of 1.6 A, and the corresponding power consumption is 11.2 W.  Table S1. Comparative analysis of the proposed SMA-based linear actuation system with a bipennate configuration vis-a-vis with existing SMA-based actuators reported in literature. '−' denotes that the data was not available in the reported article.

Case study for variable input voltage pulse
The simulation is carried out for variable voltage input condition with the pulse varying every 10 sec for a total of 60 sec as shown in Figure S1a. The actuator force was studied along with the variation of stress with temperature and variation of martensite volume fraction with temperature in the Figure S2b, S2d, and S2f respectively.  Figure S1. (a) For this simulation, a variable input voltage was considered. For the first 10 sec, 6 V was applied, then 4 V, 7 V, 3 V, 5 V, and finally 0 V for 10 sec each, (b) displays the simulation output of the temperature distribution as well as the stress-induced transition temperature of the SMA-based bipennate actuator. The temperature of the wire is directly proportional to the voltage applied across the wire. When the temperature of the wire crosses the materials austenite transition temperature in the heating phase, the modified austenite transition temperature starts to rise, and likewise happens with martensite transition temperatures when the temperature of the wire crosses the materials martensite transition temperature in the cooling phase.

Sensitivity analysis
In order to understand the influence of the physical parameters on the force output of the actuator, the sensitivity analysis study of the multi-field coupled mathematical model, as shown in Figure S6, has been carried out on the selected physical parameters to rank the parameters in the order of their influence. The affect of model parameters on the output force was observed using 2500 unique set of values which were generated using different probability distribution as shown in the Figure S3. The peak muscle force was selected as the design requirement for the study and the parametric influence of each set of variables on the force generation is shown in the scatter plot, figure Figure S4. A tornado plot is obtained from the sensitivity analysis in terms of correlation coefficients for each parameter (refer Figure 6a of main article file). V in (input voltage), K x (spring constant), α (angle of pennation), h T (convective heat transfer coefficient), l 0 (initial length of SMA), and n (number of unipennate branches) were the variables used for sensitivity analysis. Except for n, which was chosen at random from a multinomial probability distribution with equal probability for all even numbers from 4 to 24, all other parameters were chosen at random from a uniform probability distribution (the range of parameters has been illustrated in Figure S3). The simulation is performed for voltage pulse of variable input voltage condition as shown in Figure S1a. The voltage pulse consists of series of step up and down in between spanning for the total time duration of 60 sec. Figure S1b depicts the temperature response of SMA wires over time in response to a varied input voltage pulse. As shown in Figure S1b, the wire temperature rises and remains below the critical point of stress-modified austenite phase start temperature (A ′ s ) during the first 10 sec. As a result, the martensite volume fraction remains unchanged during this time because it does not reach the transformation zone. The reverse transformation occurs when the SMA temperature crosses the critical point A ′ s at t = 23.3 sec. The joule heating provided to the SMA wires from t = 20 to 30 sec is sufficient enough to allow the SMA to reach in the transformation zone. Subsequently, stress is generated in the SMA wire as a consequence of the reduction in the the martensite volume fraction as well as stroke output in the actuator is produced as shown in Figure S2a, S2c and S2e respectively. The

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SMA wire temperature keeps on increasing as the heat supply is provided followed by convective cooling during zero voltage condition. During heating, when the SMA wire temperature (T ) crosses the stress-modified austenite phase start temperature (A ′ s ), the reverse transformation from martensite to austenite phase starts to occur. At this stage, the SMA wire contracts and the muscle force is generated by the actuator. Figure S3. Multiple probabilistic distributions were used to generate 2500 unique set of the mentioned model parameters. While n was generated using multinomial distribution with each unique value having the same probability of occurrence, rest of the model parameters were generated using uniform distribution. K x ranges from 400 − 800 N/m, while n ranges from 4 − 24. Voltage (V in ) has been altered from 4 − 10 V, initial wire length (l 0 ) has been varied from 40 − 100 mm, convective heat transfer co-efficient (h T ) has been varied from 50 − 100 W/m 2 −K and pennation angle (α) has been varied from 20 − 60 • . Figure S2a and S2b show the stress induced in the SMA wire and the force generated by the actuator with respect to time. During reverse transformation (heating), when the SMA wire temperature, T < A ′ s , the rate of change of martensite volume fraction (ξ ) will be zero as given in equation (S5). Therefore, the rate of change of stress (σ ) will depend on the strain rate (ε) and the temperature gradient (Ṫ ) only as given by equation (2) (refer Constitutive equation section in the main article file). However, as the SMA wire temperature increases and crosses (A ′ s ), the austenite phase starts to form and the (ξ ) takes a value as given by equations (S1), (S3) and (S4). Hence, the stress change rate (σ ) is collectively governed byε,Ṫ andξ as given in equation (2) (refer Constitutive equation section in the main article file). This explains the change in the gradient observed in the time-dependent stress and force plots during the heating cycle as shown in Figure S2a and S2b. The same explanation holds good for the forward transformation (cooling) from austenite to martensite phase with the relation between the SMA wire temperature (T ) and the stress-modified martensite phase finish temperature (M ′ f ) as well. Figure S2d and S2f shows the variation of the stress induced in SMA wire (σ ) and the martensite volume fraction (ξ ) with respect to change in the SMA wire temperature (T ) for two actuation cycles. Figure S4. The figure depicts the distribution of the peak muscle force attributed to different combinations of model parameters. As evident from the figure, the majority of the 2500 distinct combinations picked generate forces less than 150 N, with forces larger than 150 N having a smaller but equal representation. Meanwhile, each points in the parameter space denotes one unique value of that parameter. Figure S3 shows the parameter sample space distributions for the parameters chosen for the sensitivity analysis study. For a set of six model parameters, 2500 unique set of values have been generated to study their influence on the maximum force output as shown in the Figure S4. Figure 6a (refer to main article file) shows the tornado plot of the various correlation coefficients for each parameter against the design requirement of maximum output force. It can be observed from the Figure 6a (refer to main article file) that the parameters, voltage (V in ) is directly correlated with the maximum output force, while the initial length of SMA wire (l 0 ), number of unipennate wires (n), convective heat transfer coefficient (h T ), pennation angle (α), bias spring constant (K x ) are inversely correlated with the output force. In case of direct correlation, the higher value of the correlation coefficient for the voltage (V in ) signifies that this parameter influences the force output the most while initial length of SMA wire (l 0 ) is most inversely correlated. The analysis helps in identifying and customizing the influential parameters through which the output force, stroke and efficiency of the actuator system can be tailored as per the requirement and application.

Effect of input voltage and forced convection on actuation frequency
This study elucidates the effect of external stimuli (V in and h T ) on the actuation frequency of the bipennate-based SMA actuator and the necessary parameter tuning required to achieve the actuation frequency as per the requirement and application. The simulation result reported in this paper was performed with a supply voltage (V in ) of 7 V and a forced convection condition (h T =70 W/m 2 − K). The input voltage was supplied for 10 sec during the heating cycle, followed by a 15-second cooling period. Thus, it resulted in an actuation cycle of 25 sec or a frequency of 0.04 Hz. By performing simulation, the improvement in the actuation frequency has been observed by reducing the heating cycle period by increasing the input voltage from 7V to 15V and simultaneously enhancing the forced convection, from 300 to 500 W/m 2 − K, to reduce the cooling cycle period. Figure S5d shows the input conditions of the different sets for the simulation and corresponding output in terms of the frequency, heating time, and cooling time period. Furthermore, Figure S5a and S5b show the variation in actuation force and SMA wire temperature for different sets of input voltage and convective heat transfer coefficients. Figure S5c illustrates a 303% increase in the actuation frequency upon increasing the input voltage to 15 V and forced convection to 500 W/m 2 − K simultaneously.   Figure S5.

Phase transformation equations
The time dependent martensite volume fraction can be expressed as a function of stress rate and temperature gradient in the following manner: The conditions to be satisfied for the enhanced phenomenological model 10 during phase transformation are as follows:

Reverse transformation (Martensite to Austenite)
The governing equation of reverse transformation represented by the phase change from martensite to austenite during heating condition is given by: where, A ′ s and A ′ f are stress-modified austenite phase start and finish temperatures, respectively.

Forward transformation (Austenite to Martensite)
The governing equation of forward transformation represented by the phase change from austenite to martensite during cooling condition is given by: where, M ′ s and M ′ f are stress-modified martensite phase start and finish temperatures, respectively.

Bipennate muscle stiffness equation − 1 st hierarchical level actuator
The total force generated by the SMA based bipennate muscle arrangement is given as follows: where, n is the number of unipennate branches, F m is the muscle force generated by the actuator, F f is the fiber force in the SMA wire, K x is the stiffness of the bias spring, ∆x is the stroke of the actuator, α is the angle of pennation and x 0 is the initial displacement of the bias spring to maintain the SMA wires in the pre-tension arrangement. Considering the stroke (∆x) of the muscle to be small as compared to x 0 , the mechanics equation reduces to the following form, The stiffness of the pennate muscle (k m ) can be defined as the derivative of the total muscle force generated with respect to the muscle position 11 . Upon differentiating equation (S11) with respect to muscle position, the pennate muscle stiffness (k m ) is given by: where, cos α = x/l as shown in Figure 9d (refer main article file). Upon applying the chain rule of differentiation, the equation (S12) is modified as follows: Substituting the values from equations (S14) and (S15) into equation (S13) and simplifying it further, the pennate muscle stiffness (k m ) is modified as follows: where, F f = σ A cross , E = dσ /dε, σ is the stress induced in the SMA wire, and A cross is the cross-sectional area of the SMA wire, Combining equations (S11), (S12) and (S20), it can be written as: The total displacement or stroke (∆x) of the 1 st level hierarchical actuator as a function of stress induced (σ ) and strain developed (ε) in the SMA wire is given by:

Bipennate muscle stiffness equation − 2 nd hierarchical level actuator
The total force generated by the 2 nd level SMA based bipennate muscle arrangement is given as follows: where, n 1 denotes the number of unipennate SMA wire branches, and n 2 denotes the number of secondary arms of the actuator. Considering the stroke (∆x 2 ) of the muscle to be small as compared to x 0 , the mechanics equation reduces to the following form, The stiffness of the pennate muscle (k m ) can be defined as the derivative of the total muscle force generated with respect to the position. Upon differentiating equation (S25) with respect to muscle position, the pennate muscle stiffness (k m ) is given by: Applying chain rule to equation (S26), the modified form is given as follows: As per the illustration in Figure S7 and using the similar approach as of equation (S15), the following differential and trigonometrical relations can be obtained:  Figure S7. The 2 nd level actuator is shown schematically. The angle subtended with the arm is denoted by α 1 and α 2 , where α 1 is the angle SMA wire has with the arm attached (secondary arm) and α 2 is the angle the secondary arm has with the primary arm. Similarly, n 1 represents the number of SMA wires in each arm, whereas n 2 represents the number of secondary arms connected to the primary arm. On the other hand, the projection of the SMA wire on the secondary arm is x 1 , while the projection of the secondary arm on the primary arm is x 2 .

Bipennate muscle stiffness equation − 3 rd hierarchical level actuator
The total force generated by the 3 rd level SMA based bipennate muscle arrangement is given as follows: where, n 1 represents the number of SMA wires in each 1 st level actuator-like structure, n 2 represents the number of 1 st level actuators coupled to each secondary arm, and n 3 represents the total number of secondary arms. On the other hand, α i represents the angle subtended to the arms, where i = (N = 3) (in case of 3 rd level actuator) denotes angle subtended to the primary arm and i = N − 1 denotes the angle subtended to the secondary arm and so on.

Bipennate muscle stiffness equation − N th hierarchical level actuator
Applying the principle of mathematical induction, by extrapolating the results obtained for 1 st , 2 nd , and 3 rd level actuator, and verifying for the fourth stage actuator, the total displacement or stroke (∆x N ) of the N th level hierarchical actuator as a function of stress induced (σ ) and strain developed (ε) in the SMA wire is obtained as:

Application
The bipennate muscle-based shape memory alloy actuator has broad applications ranging from building automation controls to precise drug delivery methods. Being compact, it can also act as a viable alternative to replace DC motor gear-train based conventional actuators.
• Building automation and controls: Dampers are used in industrial HVAC systems to regulate airflow. Torque ranging from 2 to 50 Nm is required by the actuators that regulate the damper. Higher torque necessitates a larger actuator, which comes at a cost. Higher torque requires an increase in actuator size with cost implications. Shape memory alloy actuators based on bipennate muscles can be used to achieve the same necessary torque with a smaller footprint and lower cost.
• Drug delivery/mixing applications: The developed SMA-based actuators can be utilized to control microvalves for cancer patients' precise medication administration systems. High force production permits drug delivery or mixing of a high viscous fluid with micrometer stroke accuracy, which is a unique feature of bipennate muscle based SMA actuators.
• Magnetic resource imaging (MRI): Magnetic resonance imaging (MRI) is particularly sensitive to electromagnetic noise produced by traditional coil-based motors, making it difficult to research neurology. Compared to its conventional equivalent, a bipennate muscle-based shape memory alloy actuator may significantly reduce the electromagnetic fields created. This aids in the reception of unaltered magnetic resonance imaging pictures free of electromagnetic noise, allowing for further analysis and extraction of neurological properties.
• Variable stiffness muscle: The current system is developed based on the bipennate muscle, which has variable stiffness. The stiffness value of the hierarchical actuator is determined by the change in SMA wire length, pennation angle, and stress generated during actuation. As a result, the present invention functions as a variable stiffness mechanism, which is advantageous for roboticists and engineers since it includes capabilities like energy storage and better human-interaction safety that are not inherent in typical kinematic linkages.
• Adaptive robotic prostheses: The interesting property of the customizable multi-stage hierarchy of the shape memory alloy-based bioinspired muscle design will also encourage researchers in the domain of biomechatronics to develop adaptive robotic prostheses.
• Antenna reconfiguration: The lab has developed an SMA based solution to control the shape of parabolic antenna reflector and thus control the antenna foot-print for space application of ISRO 12 . This reconfiguration works on a limited actuation.
where, A ′ s is the stress modified austenite start temperature and, A ′ f denotes the stress modified austenite finish temperature.